Global continuum and multiple positive solutions to a \(p\)-Laplacian boundary-value problem. (English) Zbl 1260.34045

Summary: A \(p\)-Laplacian boundary-value problem with positive nonlinearity is considered. The existence of a continuum of positive solutions emanating from \((\lambda,u)=(0,0)\) is shown, and it can be extended to \(\lambda=\infty\). Under an additional condition on the nonlinearity, it is shown that the positive solution is unique for any \(\lambda>0\); thus the continuum \(\mathcal C\) is indeed a continuous curve globally defined for all \(\lambda>0\). In addition, by the upper and lower solutions method, existence of three positive solutions is established under some conditions on the nonlinearity.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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